This equality states that the energy integral of the nonequilibrium dynamic spin structure factor at a given momentum \(q\)  is a convolution of the two-time spin correlation functions with the time-dependent profile of a probe with finite pulse duration. Note that the QFI \(f_\mathrm {Q}({q}, t)\)  is defined as \(4\langle \hat{\rho }^\mathrm {s}_{-{q}}(t) \hat{\rho }^\mathrm {s}_{{q}}(t) \rangle /N\) , which is an equal-time measurement for an instantaneous wavefunction at time \(t\)  [1]}, [2]}, [3]}. Therefore, without time-translational invariance in nonequilibrium systems, one cannot obtain the correct time evolution of the QFI by just integrating the dynamic spin structure factor.
